### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Find Positive Tangent

For triangle , what is the cotangent of angle ?

**Possible Answers:**

**Correct answer:**

The cotangent of the angle of a triangle is the adjacent side over the opposite side. The answer is

### Example Question #23 : Tangent

What is the tangent of the angle formed between the origin and the point if that angle is formed with one side of the angle beginning on the -axis and then rotating counter-clockwise to ? Round to the nearest hundredth.

**Possible Answers:**

**Correct answer:**

Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the -axis as your reference point for your angle. (Hence, it is called the "reference angle.")

Now, it is easiest to think of this like you are drawing a little triangle in the third quadrant of the Cartesian plane. It would look like:

So, the tangent of an angle is:

or, for your data, .

This is . Rounding, this is . Since is in the third quadrant, your value must be positive, as the tangent function is positive in that quadrant.

### Example Question #24 : Tangent

What is the tangent of the angle formed between the origin and the point if that angle is formed with one side of the angle beginning on the -axis and then rotating counter-clockwise to ? Round to the nearest hundredth.

**Possible Answers:**

**Correct answer:**

Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the -axis as your reference point for your angle. (Hence, it is called the "reference angle.")

Now, it is easiest to think of this like you are drawing a little triangle in the third quadrant of the Cartesian plane. It would look like:

So, the tangent of an angle is:

or, for your data, , or . Since is in the third quadrant, your value must be positive, as the tangent function is positive in this quadrant.

### Example Question #25 : Tangent

A ramp is being built at an angle of from the ground. It must cover horizontal feet. What is the length of the ramp? Round to the nearest hundredth of a foot.

**Possible Answers:**

**Correct answer:**

Based on our information, we can draw this little triangle:

Since we know that the tangent of an angle is , we can say:

This can be solved using your calculator:

or

Now, to solve for , use the Pythagorean Theorem, , where and are the legs of a triangle and is the triangle's hypotenuse. Here, , so we can substitute that in and rearrange the equation to solve for :

Substituting in the known values:

, or approximately . Rounding, this is .

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